Wong’s Equations and Charged Relativistic Particles in Non-Commutative Space

Wong’s Equations and Charged Relativistic Particles in Non-Commutative Space

Herbert BALASIN ^†, Daniel N. BLASCHKE^‡, Fran¸cois GIERES ^§ and Manfred SCHWEDA^†

† Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria

E-mail: hbalasin@tph.tuwien.ac.at, mschweda@tph.tuwien.ac.at

‡ Los Alamos National Laboratory, Theory Division, Los Alamos, NM, 87545, USA E-mail: dblaschke@lanl.gov

§ Universit´e de Lyon, Universit´e Claude Bernard Lyon 1 and CNRS/IN2P3, Institut de Physique Nucl´eaire, Bat. P. Dirac,

4 rue Enrico Fermi, F-69622-Villeurbanne, France E-mail: gieres@ipnl.in2p3.fr

Received March 02, 2014, in final form October 17, 2014; Published online October 24, 2014 http://dx.doi.org/10.3842/SIGMA.2014.099

Abstract. In analogy to Wong’s equations describing the motion of a charged relativistic point particle in the presence of an external Yang–Mills field, we discuss the motion of such a particle in non-commutative space subject to an externalU?(1) gauge field. We conclude that the latter equations are only consistent in the case of a constant field strength. This formulation, which is based on an action written in Moyal space, provides a coarser level of description than full QED on non-commutative space. The results are compared with those obtained from the different Hamiltonian approaches. Furthermore, a continuum version for Wong’s equations and for the motion of a particle in non-commutative space is derived.

Key words: non-commutative geometry; gauge field theories; Lagrangian and Hamiltonian formalism; symmetries and conservation laws

2010 Mathematics Subject Classification: 81T13; 81T75; 70S05

1 Introduction

Over the last twenty years, non-commuting spatial coordinates have appeared in various contexts in the framework of quantum gravity and superstring theories. This fact contributed to the motivation for studying classical and quantum theories with a finite or infinite number of degrees of freedom on non-commutative spaces. Different mathematical approaches have been pursued and various physical applications have been explored, e.g. see references [8,11,24,35,39] for some partial reviews. Beyond the applications, classical and quantum mechanics on non-commutative space are of interest as toy models for field theories which are more difficult to handle, in particular in the case of interactions with gauge fields. In this respect, we recall a similar situation concerning the coupling of matter to Yang–Mills fields on ordinary space: A coarser level of description for the latter theories has been proposed by Wong [45] who considered the motion of charged point particles in an external gauge field [6, 7, 9, 15, 16, 26, 27, 44].

The latter equations allow for various physical applications, e.g. to the dynamics of quarks and their interaction with gluons [26,27]. Somewhat similar equations, known as Mathisson–

Papapetrou–Dixon equations [13, 30,33] appear in general relativity for a spinning particle in

?This paper is a contribution to the Special Issue on Deformations of Space-Time and its Symmetries. The full collection is available athttp://www.emis.de/journals/SIGMA/space-time.html

curved space. In this spirit, we will consider in the present work the dynamics of a relativistic

“point” particle in non-commutative space subject to an external U_?(1) gauge field (thereby implementing a suggestion made in an earlier work [38], see also [20] for some related studies based on a first order expansion in the non-commutativity parameter).

More precisely, we are motivated by the motion in Moyal space, the latter space having been widely discussed as the arena for field theory in non-commutative spaces. By proceeding along the lines of Wong’s equations (which are discussed in Section 2), we derive a set of equations for the dynamics of the particle in Moyal space in Section3. For the description of the coupling of a particle to a gauge field, the relativistic setting is the most natural one, but our discus- sions (of Wong’s equations in commutative space or of a particle in non-commutative space) could equally well be done within the non-relativistic setting. The particular case of a con- stant field strength is the most tractable one (and the only consistent one for Wong’s equations in non-commutative space) and will be considered in some detail in Section 4. Subsequently in Section5, we briefly recall the different Hamiltonian approaches which have previously been pursued for the formulation of classical mechanics in non-commutative space and we compare the resulting equations governing the dynamics of particles coupled to an electromagnetic field.

In the appendix we present a continuum formulation of Wong’s equations on a generic space-time manifold, a formulation which readily generalizes to Moyal space.

For the motion of a relativistic point particle in four-dimensional (commutative or non- commutative) space-time, the following notations will be used. The metric tensor is given by

(η_µν)_{µ,ν∈{0,1,2,3}} = diag(1,−1,−1,−1)

and we choose the natural system of units (c ≡ 1 ≡~). The proper time τ for the particle is defined (up to an additive constant) by

dτ²=ds², with ds²≡dx^µdx_µ= (dt)²−(d~x)², (1.1) and for the massive particle we haveds² >0. From (1.1) it follows that ˙x²= 1 where ˙x² ≡x˙^µx˙µ

and ˙x^µ≡dx^µ/dτ.

2 Reminder on particles in commutative space

Abelian gauge f ield in f lat space. We consider the interaction of a charged massive rela- tivistic particle with an external electromagnetic field given by the U(1) gauge potential (A^µ).

The motion of this particle along its space-time trajectoryτ 7→x^µ(τ) is described by the action¹ S[x] =−m

Z

ds−q Z

dx^µA_µ(x(τ)) =−m Z

dτ

√

˙ x²−q

Z

dτx˙^µA_µ(x(τ)). (2.1) Here,qdenotes the conserved electric charge of the particle associated with the conserved current density (j^µ):

j^µ(y) =q Z

dτx˙^µ(τ)δ⁴(y−x(τ)), ∂_µj^µ= 0, Z

d³yj⁰(y) =q. (2.2) We note that the interaction term in the functional (2.1) may be rewritten in terms of the above current according to

q Z

dτx˙^µAµ(x(τ)) = Z

d⁴yj^µ(y)Aµ(y). (2.3)

1To be more precise, in the integral (2.1) the variableτ is viewed as a purely mathematical parameter which is only identified with proper time after deriving the equations of motion from the action. Thus, the relation ˙x²= 1 is only to be used at the latter stage.

This expression is invariant under infinitesimal gauge transformations, i.e. δ_λAµ=∂µλ, thanks to the conservation of the current:

δ_λ Z

d⁴yj^µA_µ= Z

d⁴yj^µ∂_µλ=− Z

d⁴y(∂_µj^µ)λ= 0.

We note that the parameter q which describes the coupling of the particle to the gauge field might in principle depend on the world line parameterτ: if this assumption is made,q(τ) appears under the τ-integral in (2.2), the current j^µ is no longer conserved and the coupling (2.3) is no longer gauge invariant. Thus, q is necessarily constant along the path.

Variation of the action (2.1) with respect to x^µ and substitution of the relation ˙x²= 1 leads to the familiar equation of motion

mx¨^µ=qF^µνx˙ν, with Fµν ≡∂µAν−∂νAµ. As is well known, the free particle LagrangianL_free=−m√

˙

x² which represents the first term of the action (2.1) and which is non-linear in ˙x²can be replaced by ˜Lfree = ^m₂x˙² which is linear in ˙x² since both Lagrangians yield the same equation of motion. Indeed, we will consider the latter Lagrangian in equation (2.12) and in Section5where we go over to the Hamiltonian formulation.

The aim of this work is to generalize this setting to a non-commutative space-time. Since a gauge field on non-commutative space entails a non-Abelian structure for the field strength tensorF_µν due to the star product, it is worthwhile to understand first the coupling of a spinless particle to a non-Abelian gauge field on commutative space-time.

Non-Abelian gauge f ield in f lat space. We consider a compact Lie groupG (e.g. G= SU(N) for concreteness) with generators T^a satisfying

T^a, T^b

= if^abcT^c and Tr T^aT^b

=δ^ab.

Just as for the Abelian gauge field, the source j_µ^a(x) of the non-Abelian gauge field A_µ(x) ≡ A^a_µ(x)T^a (e.g. a field theoretic expression like ¯ψγµTˆ^aψ involving a multipletψ of spinor fields) is considered to be given by the current density j_µ^a(x) of a relativistic point particle [6,7,15,16, 26,27,44,45]. Instead of an electric charge, the particle moving in an external Yang–Mills field is thus assumed to carry a color-charge or isotopic spin ~q ≡ (q^a)a=1,...,dimG which transforms under the adjoint representation of the structure group G. Henceforth, one considers the Lie algebra-valued variableq(τ)≡q^a(τ)T^awhich is assumed to beτ-dependent. The particle is then described in terms of its space-time coordinatesx^µ(τ) and its isotopic spinq(τ), i.e. it is referred to with respect to geometric space and to internal space. For the moment, we assume q(τ) to represent a given non-dynamical (auxiliary) variable and we will comment on a different point of view below. Its coupling to an external non-Abelian gauge field (Aµ) is now described by the action

S[x] =−m Z

ds− Z

dτx˙^µTr{q(τ)Aµ(x(τ))}

=−m Z

dτ

√

˙ x²−

Z

dτx˙^µq^a(τ)A^a_µ(x(τ)). (2.4)

The current density (2.2) presently generalizes to a Lie algebra-valued expression j_µ ≡ j_µ^aT^a given by

j^µ(y) = Z

dτx˙^µ(τ)q(τ)δ⁴(y−x(τ)), (2.5)

and thereby the interaction term in the functional (2.4) can be rewritten as

− Z

d⁴yTr(j^µA_µ). (2.6)

For an infinitesimal gauge transformation with Lie algebra-valued parameter λ, i.e. δ_λAµ = D_µλ≡∂_µλ−ig[A_µ, λ], we have

δ_λ Z

d⁴yTr{j^µA_µ}= Z

d⁴yTr{j^µD_µλ}=− Z

d⁴yTr{(D_µj^µ)λ}. (2.7) Thus, gauge invariance of the action (2.4) requires the current to be covariantly conserved, i.e. D_µj^µ= 0. From (2.5) we can deduce by a short calculation that

(Dµj^µ)^a(y) = Z

dτDq^a

dτ δ⁴(y−x(τ)), with Dq^a dτ ≡ dq^a

dτ −igx˙^µ[Aµ(x(τ)), q]^a, (2.8) hence j^µ is covariantly conserved if the charge q is covariantly constant along the world line:

Dq^a/dτ = 0 (subsidiary condition).

Variation of the action with respect tox^µ(and use of ˙x² = 1) yields the equations of motion mx¨^µ= Tr(qF^µν) ˙xν, where Dq^a

dτ = 0, (2.9)

and F_µν ≡ ∂_µA_ν −∂_νA_µ−ig[A_µ, A_ν]. By construction, these equations are invariant under infinitesimal gauge transformations parametrized by λ(x) since the chargeqis a non-dynamical (auxiliary) variable transforming as δ_λq(τ)≡ −ig[q(τ), λ(x(τ))] = 0 for all τ, hence

δ_λ Tr(qFµν) ˙x^ν

=−igTr(q[Fµν, λ]) ˙x^ν = igTr([q, λ]Fµν) ˙x^ν = 0, δλ

Dq dτ

=−igx˙^µ[δλAµ, q] =−ig Dλ

dτ , q

=−igD

dτ[λ, q] = 0.

Equations (2.9), which represent the Lorentz–Yang–Mills force equation for a relativistic particle in an external Yang–Mills field, are known asWong’s equations [45]. More specifically, the rela- tionDq^a/dτ = 0 may be viewed as charge transport equation and it is the geometrically natural generalization (to the charge vector (q^a)) of the constancy of the chargeq in electrodynamics.

A few remarks are in order here. We should point out that the equation of motion forx^µhad already been obtained earlier in curved space by Kerner [25], but the charge transport equation was not established in that setting. We refer to the works [6, 7, 26, 27] for a treatment of dynamics involving the Lagrangian−¹₄R

d³xTr(F^µνF_µν) of the gauge field: the latter approach yields the covariant conservation law Dµj^µ = 0 as a consequence of the equation of motion DνF^νµ =j^µ and the relation [Dµ, Dν]F^νµ = 0. For a general discussion of the issue of gauge invariance for the coupling of gauge fields to non-dynamical external sources, we refer to [34].

Equations (2.9) and their classical solutions have been investigated in the literature and applied for instance to the study of the quark gluon plasma [26]. The particular case of a constant field strength F_µν (“uniform fields”) exhibits interesting features [9] which will be commented upon in Section 4.

If one regards (q^a) as a dynamical variable which satisfies the equation of motionDq^a/dτ = 0 and which transforms under gauge variations with the adjoint representation, i.e.

δλq(τ) =−ig[q(τ), λ(x(τ))], (2.10) then equations (2.9) are obviously gauge invariant since Fµν also transforms with the adjoint representation². However, we emphasize that we started from the action (2.4) to obtain the equation of motion ofx^µ, the one ofq following from the requirement of gauge invariance of the

2Ifj^µis assumed to transform covariantly, then gauge invariance of the action (2.6) implies∂µj^µ= 0 as has already been noticed in reference [36].

initial action³. An action which yields the equations (2.9) as equations of motion of both x^µ andq has been constructed in the non-relativistic setting in reference [28]. The relativistic gene- ralization of this approach proceeds as follows. (For simplicity, we put the coupling constant g equal to one.) One introduces a Lie algebra-valued variable Λ(τ) ≡ Λ^a(τ)T^a where the func- tions Λ^a(τ) are Grassmann odd, the charge q being defined as an expression which is bilinear in Λ (i.e. a description which is familiar for the spin):

q ≡ −1

2[Λ,Λ]₊, i.e. q^a=−i

2f^abcΛ^bΛ^c. (2.11)

The Lagrangian

L(x,x,˙ Λ,Λ)˙ ≡ m 2x˙²+ i

2Tr

ΛDΛ dτ

= m 2x˙²+ i

2Tr(Λ ˙Λ) + ˙x^µTr(qA_µ), (2.12) is invariant under gauge transformations for which δ_λAµ=Dµλand

δ_λΛ(τ) =−i[Λ(τ), λ(x(τ))],

which implies the transformation law (2.10). Moreover, the Lagrangian leads to the following equations of motion for x^µ and Λ^a:

mx¨µ= ˙x^νTr{q(∂_µAν−∂νAµ)}+ Tr( ˙qAµ), 0 = DΛ^a

dτ . (2.13)

From (2.11) it follows that ^Dq_dτ =− Λ,^DΛ_dτ

+, hence (2.13) implies that ^Dq_dτ^a = 0, i.e. the chargeq is covariantly conserved. Substitution of the latter result into the first of equations (2.13) yields the equation of motion m¨x^µ = Tr(qF^µν) ˙xν. We note that the Hamiltonian associated to the Lagrangian (2.12) reads

H = 1

2m[p^µ−Tr(qA^µ)][p_µ−Tr(qA_µ)],

orH = ^m₂x˙² if expressed in terms of the velocity. The Poisson brackets {x^µ, pν}=δ^µ_ν, {Λ_a,Λb}=−iδ_ab,

which imply the non-Abelian algebra of charges {q_a, qb}=fabcqc,

again allow us to recover all previous equations of motion from the evolution equation of dynami- cal variablesF, i.e. from ˙F ={F, H}. In terms of the kinematical momentum Π_µ≡pµ−Tr(qA_µ), the Hamiltonian readsH = _2m¹ Π² and the Poisson brackets take the form

{x^µ,Π_ν}=δ^µ_ν, {Π_µ,Π_ν}= Tr(qF_µν), {q_a, q_b}=f_abcq_c.

The dynamical variable Λ which allowed for the Lagrangian formulation is well hidden in the latter equations.

Continuum formulation of the dynamics. The field strengthFµν manifests itself physi- cally by the force field, i.e. by an exchange of energy and momentum between the charge carrier

3In fact, if the chargeq is treated as a dynamical variable in the action (2.4), then it amounts to a Lagrange multiplier leading to the equation of motion ˙x^µA^a_µ(x(τ)) = 0 which is not gauge invariant. We thank the anonymous referee for drawing our attention to this point.

and the field [40]. In the framework of field theory, the physical entities are described by lo- cal fields, i.e. one has a continuum formulation. In order to obtain such a formulation for the particle’s equations of motion (2.9), we have to integrate these relations over the variable τ with a delta function concentrated on the particle’s trajectory. The resulting expressions then involve the current density j^aµ(y) defined in equation (2.5) as well as the energy-momentum tensor (density) of the point particle which is given by (see Appendix A)

T^µν(y) = Z

dτ mx˙^µx˙^νδ⁴(y−x(τ)). (2.14)

More explicitly, by using

˙

x^ν∂_ν^yδ⁴(y−x(τ)) =−x˙^ν∂_ν^xδ⁴(y−x(τ)) =− d

dτδ⁴(y−x(τ)), (2.15)

we have

∂_ν^yT^νµ(y) = Z

dτ mx˙^µx˙^ν∂_ν^yδ⁴(y−x(τ)) =− Z

dτ mx˙^µ d

dτδ⁴(y−x(τ))

= Z

dτ m¨x^µδ⁴(y−x(τ)),

and substitution of the particle’s equation of motion m¨x^µ= Tr(qF^µν) ˙x_ν then yields

∂_ν^yT^νµ(y) = Z

dτx˙νq^aF_a^µν(x(τ))δ⁴(y−x(τ)) =F_a^µν(y) Z

dτx˙νq^aδ⁴(y−x(τ))

=F_a^µν(y)j_ν^a(y).

The continuum version of Wong’s equations (2.9) thus reads

∂νT^νµ= Tr(F^µνjν), where Dµj^µ= 0. (2.16)

These equations describe the exchange of energy and momentum between the field F^µν and the current j^µ (i.e. the matter). They admit an obvious generalization to Moyal space, see equation (3.8) below. In Appendix A, we show that they also admit a natural extension to curved space (endowed with a metric tensor (gµν(x))). Moreover, we will prove there that they have to hold for arbitrary dynamical matter fields φ whose dynamics is described by a generic actionS[φ;g_µν, A^a_µ] which is invariant under both gauge transformations and general coordinate transformations (gµν and A^a_µ representing fixed external fields). We note that the expectation values of equations (2.16) viewed as operatorial relations in quantum field theory imply Wong’s classical equations of motion for sufficiently localized, quantum “wave-packet” states [9].

Curved space. Finally, we also point out that equations which are somewhat similar to Wong’s equations appear in general relativity for a spinning particle in curved space, for which case the contraction of the Riemann tensor R^α_βµν with the spin tensorS^µν plays a role which is similar to the field strengthF^µν in Yang–Mills theories. The explicit form of these equations of motion, which are known as theMathisson–Papapetrou–Dixon equations [13,30,33], is given by

∇

dτ(mu^α) +1

2S^µνu^σR^α_σµν = 0, ∇S^αβ dτ = 0,

where _dτ^∇ denotes the covariant derivative along the trajectory and (u^µ) is the particle’s four- velocity.

3 Lagrangian approach to particles in NC space

Moyal space and distributions. We consider four dimensional Moyal space, i.e. we assume that the space-time coordinates fulfill a Heisenberg-type algebra (for a review see [8,35,39] and references therein). Thus, the star product of functions is defined by

(f ? g)(x)≡ e^{2iθµν∂µx∂νy}f(x)g(y) x=y,

where the parameters θ^µν = −θ^νµ are constant, and their star commutator is defined by [f ^?, g]≡f ? g−g ? f, which implies that [x^{µ ?}, x^ν] = iθ^µν. In the sequel we will repeatedly use the following fundamental properties of the star product:

Z

d⁴xf ? g= Z

d⁴xf ·g, Z

d⁴xf ? g ? h= Z

d⁴xh ? f ? g. (3.1)

An important point to note is that in Moyal space, the integral R

d⁴xplays the role of a trace⁴, and hence equations of motion must always be derived from the action rather than the Lag- rangian. For a detailed discussion of the algebras of functions and of distributions on Moyal space in the context of non-commutative spaces and of quantum mechanics in phase space, we refer to [43] and [21, 42], respectively. Here, we only note that the star product of the delta distribution δ_y (with support in y) with a function ψ may be defined by application to a test functionϕ:

hδ_y? ψ, ϕi ≡ Z

d⁴x(δy? ψ)(x)ϕ(x) = Z

d⁴x(δy? ψ ? ϕ)(x) = Z

d⁴xδy(x)(ψ ? ϕ)(x)

= (ψ ? ϕ)(y).

Hence, the action of the distribution δ_y? ψ on the test function ϕ is equal to the action of the distributionδy on the test function ψ ? ϕ. Similarly, we find

hψ ? δ_y, ϕi ≡ Z

d⁴x(ψ ? δy)(x)ϕ(x) = Z

d⁴x(ψ ? δy ? ϕ)(x) = Z

d⁴x(δy? ϕ ? ψ)(x)

= (ϕ ? ψ)(y).

The following considerations hold for an arbitrary antisymmetric matrix (θ^µν), but for the physical applications it is preferable to assume thatθ^µ0= 0, i.e. assume the time to be commut- ing with the spatial coordinates. This choice is motivated by the fact that the parameters θ^ij have close analogies with a constant magnetic field both from the algebraic and dynamical points of view [11], and by the fact that a non-commuting time leads to problems with time-ordering in quantum field theory [5].

Charged particle in Moyal space. Since aU_?(1) gauge field (A^µ) on Moyal space entails a non-Abelian structure for the field strength tensor (Fµν) due to the star product [8,35,39],

F_µν ≡∂_µA_ν −∂_νA_µ−ig[A_µ^?, A_ν],

one expects that the treatment of a source for this gauge field as given by a “point” particle should allow for a description of such particles on non-commutative space which is quite similar to the one found by Wong in the case of Yang–Mills theory. If the matter content in field theory is given by a spinor fieldψ, then the interaction term with the gauge field reads

Z

d⁴yJ^µ? Aµ, with J^µ≡gψγ¯ ^µ? ψ,

4This is best seen by employing the Weyl map from operators to functions, and is also the reason for the cyclicity property (3.1). Furthermore, when considering gauge fields only the action is gauge invariant, not the Lagrangian.

i.e. it involves two star products. By virtue of the properties (3.1) one of these star products can be dropped under the integral, but not both of them. If we consider the particle limit (i.e. J^µ representing the current density of the particle), it is judicious to maintain the star product betweenJ^µandAµso as not to hide the non-commutative nature of the underlying space (over which we integrate) and to allow for the use of the cyclic invariance property (3.1) later on in our derivation. In fact, the pairinghJ, Ai ≡R

d⁴yJ^µ? A_µ represents the analogue of the pairing hj, Ai ≡ R

d⁴yTr(j^µAµ) in Yang–Mills theory on commutative space. Accordingly, we will require the action for the particle to be invariant under non-commutative gauge transformations defined at the infinitesimal level by δ_λA_µ =D_µλ≡∂_µλ−ig[A_µ ^?, λ] where the parameter λ is an arbitrary function. As was done for Wong’s equations, we assume that the charge q of the relativistic particle in non-commutative space depends on the parameter τ parametrizing the particle’s world line and that it represents a given non-dynamical variable. The coupling of this particle to an externalU?(1) gauge field (A^µ) can be described by the action⁵

S[x]≡S_free[x]−Sint[x]≡ −m Z

dτ

√

˙ x²−

Z

d⁴yJ^µAµ, (3.2)

where

J^µ(y)≡ Z

dτ q(τ) ˙x^µ(τ)δ⁴(y−x(τ)). (3.3)

Keeping in mind the formulation of field theory on non-commutative space [8, 35, 39], we will argue directly with the action rather than the Lagrangian function and require this ac- tion to be invariant under non-commutative gauge transformations. For such an infinitesimal transformation we have

δ_λ Z

d⁴yJ^µA_µ= Z

d⁴yJ^µD_µλ=− Z

d⁴y(D_µJ^µ)? λ.

Hence, invariance of the action (3.2) under non-commutative gauge transformations requires the current to be covariantly conserved, i.e. DµJ^µ = 0. By virtue of equation (3.3) we now infer that

0= (D^! _µJ^µ)(y) = Z

dτ qx˙^µD^y_µδ⁴(y−x(τ))

= Z

dτ qx˙^µ

∂_µ^yδ⁴(y−x(τ))−ig[A_µ(y)^?, δ⁴(y−x(τ))] . (3.4) Equation (2.15) entails that the first term in the last line can be rewritten as

Z

dτ qx˙^µ∂_µ^yδ⁴(y−x(τ)) =− Z

dτ q d

dτδ⁴(y−x(τ)) = Z

dτdq

dτδ⁴(y−x(τ)).

From condition (3.4) it thus follows that the chargeq has to becovariantly conserved along the world line in the sense that

0 = Z

dτDq

dτδ⁴(y−x(τ))≡ Z

dτ dq

dτδ⁴(y−x(τ))−igqx˙^µ[Aµ(y)^?, δ⁴(y−x(τ))]

. (3.5) For later reference, we note that this relation yields the following equality after star multiplication with Aν(y)δy^ν and integration overy:

Z d⁴y

Z

dτ δx^ν(τ)dq

dτδ⁴(y−x(τ))? Aν(y)

5We note that the coupling of U?(1) gauge fields to external currents has also been addressed in the recent work [4].

= ig Z

d⁴y Z

dτ δx^ν(τ)qx˙^µ[Aµ(y)^?, δ⁴(y−x(τ))]? Aν(y)

=−ig Z

d⁴y Z

dτ δx^ν(τ)qx˙^µδ⁴(y−x(τ))?[A_µ(y)^?, A_ν(y)]. (3.6) In order to derive the equation of motion for the particle determined by the action (3.2), we vary the latter with respect to x^µ. The variation of S_free being the same as in commutative space, we only work out the variation of the interaction partSint, all star products being viewed as functions of the variable y:

δS_int =δ Z

d⁴y Z

dτ

˙

x^µqδ⁴(y−x(τ))? A_µ(y)

= Z

d⁴y Z

dτ

δx˙^µqδ⁴(y−x(τ))? Aµ(y) + ˙x^µqδ

δ⁴(y−x(τ))

? Aµ(y)

= Z

d⁴y Z

dτ

d(δx^µ)

dτ qδ⁴(y−x(τ))? A_µ(y) + ˙x^µq(δx^ν)∂_ν^xδ⁴(y−x(τ))? A_µ(y)

= Z

d⁴y Z

dτ(δx^ν)

− d dτ

qδ⁴(y−x(τ))

? Aν(y) + ˙x^µqδ⁴(y−x)? ∂_ν^yAµ(y)

. By virtue of the product rule, the first term in the last line yields two terms, one involving

dq

dτ which can be rewritten using relation (3.6), and one involving _dτ^dδ⁴(y−x(τ)) which can be rewritten using (2.15):

− Z

d⁴y Z

dτ(δx^ν)q d

dτδ⁴(y−x(τ))? A_ν(y)

= Z

d⁴y Z

dτ(δx^ν)qx˙^µ

∂_µ^yδ⁴(y−x(τ))

? Aν(y)

=− Z

d⁴y Z

dτ(δx^ν)qx˙^µδ⁴(y−x(τ))? ∂_µ^yAν(y).

Hence, we arrive at δS_int =

Z d⁴y

Z

dτ(δx^ν) ˙x^µqδ⁴(y−x(τ))?

∂_ν^yA_µ(y)−∂_µ^yA_ν(y) + ig[A_µ(y)^?, A_ν(y)]

= Z

d⁴y Z

dτ(δx^ν) ˙x^µqδ⁴(y−x(τ))? Fνµ(y) = Z

dτ(δx^ν) ˙x^µqFνµ(x(τ)).

Note that it is the constraint equation (3.6) following from (3.5) that yields the terms which are quadratic in the gauge field. This is very much the same mechanism as in the commutative space calculation which leads to Wong’s equations, cf. (2.4)–(2.9).

In conclusion, we obtain the followingequation of motion for the charged relativistic particle in non-commutative space:

mx¨^µ=qF^µνx˙ν, with Fµν ≡∂µAν−∂νAµ−ig[Aµ?, Aν]. (3.7) We note that the antisymmetry ofF_µν with respect to its indices implies

0 = ¨x^µx˙_µ= 1 2

dx˙² dτ ,

which is consistent with ˙x² = 1. Consistency of the equation of motion (3.7) requires its gauge invariance, i.e. the gauge invariance of its right-hand-side. Since

δ_λ(qF^µνx˙ν) =q(δ_λF^µν) ˙xν =−igq[F^{µν ?}, λ] ˙xν =gqθ^ρσ(∂ρF^µν)(∂σλ) ˙xν +O θ² ,

the gauge invariance only holds for constant field strengths by contrast to the case of Wong’s equation for Yang–Mills fields in commutative space. This difference can be traced back to the fact that the analogue of the trace in Yang–Mills theory is given in Moyal space by the integralR

d⁴x: since such an integral does not occur in the differential equation (3.7), the gauge invariance is only realized for constant F_µν. We will further discuss the latter fields and the comparison with Yang–Mills theory in Section 4 where we will also consider the subsidiary condition for the chargeq. Here we only note the following points in this respect. The equation of motion (3.7) taken for itself is consistent for any constant values of q and F_µν. Furthermore in non-commutative space the auxiliary variable q cannot be rendered dynamical with a non- trivial gauge transformation law (2.10) for q(τ) since the star commutator ofq(τ) with a gauge transformation parameter λ(x(τ)) vanishes.

In summary, the coupling of a relativistic particle to a gauge field (A^µ) is described in general by the Lagrangian

L(x,x) =˙ −m

√

˙

x²−qAµx˙^µ, or L(x,x) =˙ m

2x˙²+qAµx˙^µ.

The non-commutativity of space-time can be implemented in the Lagrangian framework by rewriting the interaction term of the action as an integralR

d⁴y(J^µ?A_µ)(y) where the currentJ^µ is defined by (3.3) and by requiring this action to be invariant under non-commutative gauge transformations. The resulting equation of motion (3.7) is only gauge invariant for constant field strengths.

Continuum formulation of the dynamics. By following the same lines of arguments as for the Lorentz–Yang–Mills force equation (see equations (2.14)–(2.16)), we can obtain a continuum version of the equation of motion (3.7) by multiplying this equation withδ⁴(y−x(τ))ϕ(y), where ϕ(y) is a suitable test function, and integrating over τ and over y. More explicitly, by starting from the energy-momentum tensor (2.14) for the point particle and using relation (2.15), we get

Z

d⁴y(∂νT^νµ)(y)ϕ(y) = Z

d⁴y Z

dτ mx˙^µx˙^ν∂_ν^yδ⁴(y−x(τ))ϕ(y)

=− Z

d⁴y Z

dτ mx˙^µ d

dτδ⁴(y−x(τ))ϕ(y) = Z

d⁴y Z

dτ m¨x^µδ⁴(y−x(τ))ϕ(y).

Substitution of equation (3.7) then yields the expression Z

d⁴y Z

dτ qF^µν(x(τ)) ˙xνδ⁴(y−x(τ))ϕ(y) = Z

d⁴y Z

dτ qF^µν(y) ˙xνδ⁴(y−x(τ))ϕ(y)

= Z

d⁴y(F^µνJ_ν)(y)ϕ(y),

where we considered the current density (3.3) in the last line. Thus, we have the result

∂_νT^νµ=F^µνJ_ν, where D_µJ^µ= 0, (3.8)

and these relations are completely analogous to the continuum equations (2.16) which correspond to Wong’s equations (apart from the fact that the invariance of (3.8) under non-commutative gauge transformations requires the field strength F^µν to be constant).

4 Case of a constant f ield strength

We will now discuss the dynamics of charged particles coupled to a constant field strength on the basis of the results obtained in Sections 2 and 3. Indeed this case represents a mathematically tractable and physically interesting application of the general formalism.

We successively discuss the case of non-Abelian Yang–Mills theory on Minkowski space and the case of a U?(1) gauge field on Moyal space while emphasizing the differences that exist for constant field strengths.

4.1 Wong’s equations in commutative space

The case of a “uniform field strength” in Yang–Mills theory has been addressed some time ago by the authors of reference [9], see also [19] for related points. Since the Yang–Mills field strength F_µν(x) ≡ F_µν^a (x)T^a is not gauge invariant, but transforms under finite gauge transformations as F_µν⁰ =U⁻¹FµνU (with U(x) ∈G= structure group), one has to specify first what is meant by a constant field. The field F_µν is said to be uniform if the gauge field A_µ(x) ≡ A^a_µ(x)T^a at a point x can be related by a gauge transformation to the gauge field A_µ(y) at any other pointy. More precisely, for a space-time translation parametrized bya∈R⁴, there exists a gauge transformation x7→U(x;a)∈Gsuch that

Aµ(x+a) =U⁻¹(x;a)Aµ(x)U(x;a) + iU⁻¹(x;a)∂µU(x;a), and thereby

Fµν(x+a) =U⁻¹(x;a)Fµν(x)U(x;a).

In this case a gauge may be chosen in which all components of F_µν are constant becauseF_µν at the point x can be made equal to its value at some arbitrary point y by transforming it by an appropriate gauge group element.

Since the field strengthF_µν ≡∂_µA_ν −∂_νA_µ+ i[A_µ, A_ν] contains two terms, namely ∂_µA_ν −

∂νAµ which has the Abelian form, and [Aµ, Aν] which does not involve derivatives, a constant non-zero field strengthFµν can be obtained either from a linear gauge potential (i.e. an Abelian- like gauge field),

A_µ=−1

2F_µνx^ν, ∂_µA_ν−∂_νA_µ=F_µν = const, [A_µ, A_ν] = 0, or from a constant non-Abelian-like gauge potential,

A_µ= const, ∂_µA_ν = 0, i[A_µ, A_ν] =F_µν = const.

In fact [9], these two types of potentials exhaust all possibilities for a constant field strength.

It has been shown for the structure group SU(2) that the two types of gauge potentials leading to a same constant field strength are gauge inequivalent and result in physically different be- havior when matter interacts with them, e.g. the solutions of Wong’s equations have completely different properties in both cases. An explicit example for G = SU(2) is given by a constant magnetic field inz-direction [19]: letσ_k(withk= 1,2,3) denote the Pauli matrices and suppose

F0i = 0, Fij =εijkBk, with (Bk)k=1,2,3= (0,0,2σ3).

This constant field strength derives from the linear Abelian-like potential A~ ≡(A_k)_k=1,2,3=−1

2~x∧B~ = (−y, x,0)σ₃,

or from a constant non-Abelian-like potentialA~= (−σ₂, σ₁,0).

4.2 Wong’s equations in non-commutative space

The field strength associated to aU?(1) gauge field (Aµ) on Moyal space reads Fµν ≡∂µAν −∂νAµ−ig[Aµ?, Aν] =∂µAν−∂νAµ+gθ^ρσ(∂ρAµ)(∂σAν) +O θ²

.

Due to the derivatives appearing in the star commutator, the Abelian-like term ∂µAν −∂νAµ

and the non-Abelian-like term −ig[A_µ ^?, Aν] cannot vanish independently of each other: the

non-commutative field strength Fµν can only be constant for a linear Abelian-like potential.

More precisely, for Aµ=−1

2B¯µνx^ν, (4.1)

where the coefficients ¯B_µν ≡ −B¯_νµ are constant, we obtain Fµν = ¯Bµν− g

4B¯µρθ^ρσB¯σν. (4.2)

This field strength is constant, but dependent on the non-commutativity parameters θ^µν. If we interpret ¯Bµν = ∂µAν −∂νAµ as the physical field strength, then (4.2) means that the non- commutativity parametersθ^µν modify in general the trajectories of the particle as compared to its motion in commutative space⁶.

Since the field strength transforms under a finite gauge transformation U(x) ∈ U?(1) as F_µν =U⁻¹?F_µν?U, a constant fieldF_µνis gauge invariant. Thus, for constant field strengths the situation is quite different forU_?(1) gauge fields and for non-Abelian gauge fields on Minkowski space despite the fact that we encounter the same structure for the gauge transformations, the field strength and the action functional in both cases.

Let us again come back to the expression (4.1) for the gauge field. Substitution of this expression into the subsidiary condition (3.5) yields

0 = Z

dτ

˙

qδ⁴(y−x(τ))−igqx˙^µ[Aµ(y)^?, δ⁴(y−x(τ))]

= Z

dτ

˙

qδ⁴(y−x(τ))−g

2qx˙^µB¯_µρθ^ρσ∂_σ^yδ⁴(y−x(τ)) . (4.3) If the matrix ( ¯B_µν) is the inverse of the matrix (θ^µν), i.e. ¯B_µρθ^ρσ =δ_µ^σ, thenF_µν = (1−^g₄) ¯B_µν and, by virtue of (2.15) and an integration by parts, condition (4.3) takes the form

0 = Z

dτqδ˙ ⁴(y−x(τ))n 1−g

2 o

.

The latter relation is obviously satisfied for a constant q. In this case, the equation of mo- tion (3.7) for the particle in non-commutative space, i.e. mx¨^µ=qF^µνx˙ν, has the same form as the one of an electrically charged particle in ordinary space. This result is analogous to the one obtained for a constant magnetic field in x³-direction within the Hamiltonian approaches, see equations (5.10) and (5.13) below.

5 Hamiltonian approaches to particles in NC space

To start with, we briefly review the Hamiltonian approaches in commutative space before con- sidering the generalization to the non-commutative setting. In the latter setting, we will notice that various approaches yield different results since several expressions which coincide in com- mutative space no longer agree.

6We note that the latter dependence on the non-commutativity parameters can be eliminated mathematically if one assumes that ¯Bµν depends in a specific way on the parametersθ^µν and someθ-independent constantsBµν. To illustrate this point [11], we assume that the only non-vanishing components of ¯Bµν and θ^µν are as follows:

B¯12=−B¯21 ≡B, θ¯ ¹² =−θ²¹ ≡θ, i.e.F12= ¯B(1 +^g₄θB) =¯ −F21. If ¯B depends onθ and on aθ-independent constantB according to

B¯= ¯B(B;θ)≡ 2 gθ

p1 +gθB−1

=B 1−g

4θB

+O θ² ,

then relation (4.2) implies that the non-commutative field strengthF12 is aθ-independent constant: F12=B.

5.1 Reminder on the Poisson bracket approach

The Hamiltonian formulation of relativistic (as well as non-relativistic) mechanics is based on two inputs (e.g. see reference [29]): a Hamiltonian function and a Poisson structure (or equivalently a symplectic structure). If one starts from the Lagrangian formulation, the Hamiltonian function is obtained from the Lagrange function by a Legendre transformation. E.g. the Lagrangian L(x,x) =˙ ^m₂x˙²+eAµx˙^µ (involving the constant chargee) yields theHamiltonian

H(x, p) = 1

2m(p−eA)² = 1

2m(p^µ−eA^µ)(pµ−eAµ). (5.1)

The trajectories in phase space are parametrized by τ 7→ (x(τ), p(τ)) where τ denotes a real variable to be identified with proper time after the equations of motion have been derived. The Poisson brackets {·,·} of the phase space variables x^µ, p^µ are chosen in such a way that the evolution equation ˙F ={F, H} (where ˙F ≡dF/dτ) yields the Lagrangian equation of motion for x^µ, though written as a system of first order differential equations. For instance, if we consider the usual form of the Poisson brackets, i.e. the canonical Poisson brackets

{x^µ, x^ν}= 0, {p^µ, p^ν}= 0, {x^µ, p^ν}=η^µν, (5.2) then substitution of F =x^µ and F =p^µ into ˙F = {F, H} (with H given by (5.1)) yields the system of equations

mx˙^µ=p^µ−eA^µ, mp˙^µ=e(p_ν−eA_ν)∂^µA^ν, from which we conclude that

mx¨^µ= ˙p^µ−eA˙^µ= e

m(p_ν −eA_ν)

| {z }

=ex˙ν

∂^µA^ν −ex˙^ν∂_νA^µ=ex˙_ν ∂^µA^ν −∂^νA^µ

≡ef^µνx˙_ν.

This equation coincides with the Euler–Lagrange equation forx^µfollowing from the Lagrangian L(x,x) =˙ ^m₂x˙²+eAµx˙^µ.

Concerning the gauge invariance, we emphasize a result [40] which does not seem to be very well known. The Hamiltonian (5.1) is not invariant under a gauge transformation Aµ → Aµ+∂µλwhich is quite intriguing. However, it is invariant if this transformation is combined with the phase space transformation (x^µ, p^µ) → (x^µ, p^µ+e∂^µλ), the latter being a canonical transformation since it preserves the fundamental Poisson brackets (5.2). Indeed, under this combined transformation the kinematical momentum Π^µ≡p^µ−eA^µ(which coincides withmx˙^µ) is invariant.

We note that the Hamiltonian (5.1) can be rewritten in terms of the variable Π_µ≡p_µ−eA_µ asH≡ _2m¹ Π². TherebyH has the form of a free particle Hamiltonian, but the Poisson brackets are now modified: from Π_µ=p_µ−eA_µ and (5.2) it follows that

{x^µ, x^ν}= 0, {Π^µ,Π^ν}=ef^µν(x), {x^µ,Π^ν}=η^µν, (5.3) with f^µν ≡∂^µA^ν −∂^νA^µ. Since the electromagnetic field strength f^µν is gauge invariant, the latter invariance is manifestly realized in this formulation.

In summary, the coupling of a charged particle to an electromagnetic field can either be described by the canonical Poisson brackets (5.2) and the minimally coupled Hamiltonian (5.1) or by introducing the field strength into the Poisson brackets (as a non-commutativity of the momenta) and considering a Hamiltonian which has the form of a free particle Hamiltonian.

5.2 Reminder on the symplectic form approach

If we gather all phase space variables into a vector ξ~ ≡ (ξ^I) ≡ (x⁰, . . . , x³, p⁰, . . . , p³), the fundamental Poisson brackets (5.2) read

ξ^I, ξ^J = Ω^IJ, with Ω^IJ

≡

0 η^µν

−η^µν 0

.

The inverse of the Poisson matrix Ω ≡(Ω^IJ) is the matrix with entries ω_IJ ≡(Ω⁻¹)_IJ which defines the symplectic 2-form

ω≡ 1 2

X

I,J

ω_IJdξ^I∧dξ^J =dp^µ∧dx_µ, (5.4)

e.g. see reference [29] for mathematical details. The Hamiltonian equations of motion can be written as

ξ˙^I =

ξ^I, H = Ω^KJ∂Kξ^I∂JH, i.e. ξ˙^I= Ω^IJ∂JH, or equivalently asω_IJξ˙^J =∂_IH.

In terms of the phase space variables (x^µ,Π^µ) appearing in the non-canonical Poisson brac- kets (5.3), the symplectic 2-form (5.4) reads

ω=dΠ^µ∧dx_µ+1

2ef_µνdx^µ∧dx^ν.

This formulation of the electromagnetic interaction based on the symplectic 2-form and the evolution equation ω_IJξ˙^J =∂_IH goes back to the seminal work of Souriau [37].

5.3 Standard (Poisson bracket) approach to NC space-time

In order to introduce a non-commutativity for the configuration space, one generally starts from a function H on phase space to which one refers as the Hamiltonian without any ref- erence to a Lagrangian, e.g. we can consider the function H given in equation (5.1). The non-commutativity of space-time is then implemented by virtue of the Poisson brackets

{x^µ, x^ν}=θ^µν, {p^µ, p^ν}= 0, {x^µ, p^ν}=η^µν, (5.5) whereθ^µν =−θ^νµ is again assumed to be constant. (For an overview of the description of non- relativistic charged particles in non-commutative space we refer to [3, 11, 24], the pioneering work being [17,18], see also [1,2,31,32] for some subsequent early work. We also mention that dynamical systems in non-commutative space can be constructed by applying Dirac’s treatment of constrained Hamiltonian systems to an appropriate action functional, see [12] and references therein.)

As in the commutative setting, we gather all phase space variables into a vectorξ~≡(ξ^I) ≡ (x⁰, . . . , x³, p⁰, . . . , p³), the fundamental Poisson brackets (5.5) being now given by

{ξ^I, ξ^J}= Ω^IJ, with (Ω^IJ)≡

θ^µν η^µν

−η^µν 0

.

Quite generally, the Poisson bracket of two arbitrary functionsF,Gon phase space reads {F, G} ≡X

I,J

Ω^IJ∂_IF ∂_JG=θ^µν ∂F

∂x^µ

∂G

∂x^ν + ∂F

∂x^µ

∂G

∂p_µ − ∂F

∂p^µ

∂G

∂x_µ.

Substitution of F = x^µ and F = p^µ into ˙F = {F, H} (with H given by (5.1)) yields the system of equations

mx˙^µ= (pν−eAν)(η^µν−eθ^µρ∂ρA^ν),

mp˙^µ=e(p_ν −eA_ν)∂^µA^ν. (5.6)

In the present case, the phase space transformation (x^µ, p^µ)→(x^µ, p^µ+e∂^µλ) does not represent a canonical transformation since it does not preserve the Poisson brackets (5.5) ifθ^µν 6= 0. Hence, the resulting Hamiltonian equations of motion (5.6) are not gauge invariant, as has already been pointed out in reference [11] by considering different gauges.

In the next two subsections, we recall how this problem can be overcome for the particular case of a constant field strength as well as more generally, and we compare with the results obtained in Section 3 from the action involving star products. Here, we only note that a non- Abelian structure of the field strength is hidden in equation (5.6). To illustrate this point, we consider the particular case where the only non-vanishing components of θ^µν are θ^ij = ε^ijθ (with i, j ∈ {1,2} and ε¹² ≡ −ε²¹ ≡ 1) and where the only non-vanishing components of A^µ areAⁱ(x¹, x²) (withi∈ {1,2}). For this situation which describes a time-independent magnetic field perpendicular to thex¹x²-plane, the first of equations (5.6) yields

mx˙i = (pk−eAk)(δik−eθεij∂jAk), and implies

md

dt(x_i+eθε_ijA_j) = (1 +eF₁₂)(p_i−eA_i) (5.7) with

F₁₂≡∂1A2−∂2A1+e{A₁, A2}=∂1A2−∂2A1+eθ^ρσ(∂ρA1)(∂σA2). (5.8) Thus, we find a non-Abelian structure for the generalized field strength, but in the present approach the field F_µν is only linear in the non-commutativity parameters in contrast to the fieldFµν in (3.7) which involves the star commutator

−i[A_µ^?, A_ν] =θ^ρσ(∂_ρA_µ)(∂_σA_ν) +O θ² .

If the gauge potential is linear inx, the field strengthsF_µνandFµνas defined by equations (5.8) and (3.7), respectively, coincide with each other (if one identifies the coupling constantgwithe).

To conclude, we note that (5.7) can be solved for p_i−eA_i in terms of mx˙_i: the system of first order differential equations (5.6) can then be written as a second order equation for x^µ, but the resulting equations of motion are not gauge invariant and they cannot be derived from a Lagrangian [3].

5.4 Standard approach to NC space-time continued

The reasoning presented concerning the brackets (5.3) suggests to consider a Hamiltonian which has a free form and to introduce a field strengthB^µν(x) as a non-commutativity of the momenta, i.e. consider phase space variables (x^µ, p^µ) satisfying the non-canonical Poisson algebra

{x^µ, x^ν}=θ^µν, {p^µ, p^ν}=eB^µν, {x^µ, p^ν}=η^µν, (5.9) with θ^µν constant. As pointed out in reference [22], the Jacobi identities for the algebra (5.9) are only satisfied if the field strength is constant:

{x^µ,{p^ν, p^λ}}+{cyclic permutations of µ,ν,λ}=eθ^µρ∂ρB^νλ.